University of Rhode Island DigitalCommons@URI
Open Access Master's Theses
2015
SCHEDULE OPTIMIZATION WITH FLEXIBLE DURATIONS
Kevin R. Roy University of Rhode Island, [email protected]
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SCHEDULE OPTIMIZATION WITH FLEXIBLE
DURATIONS
BY
KEVIN R. ROY
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
SYSTEMS ENGINEERING
UNIVERSITY OF RHODE ISLAND
2015
MASTER OF SCIENCE
OF
KEVIN R. ROY
APPROVED:
Thesis Committee:
Major Professor Manbir Sodhi
David Taggart
Cenk Undey
Nasser Zawia DEAN OF THE GRADUATE SCHOOL
UNIVERSITY OF RHODE ISLAND 2015
ABSTRACT
Cost, quality, and time are the pillars of a popular and well known business model. What is also well known about this model is that it is very difficult to be the industry leader in all three categories. This is because these categories are naturally interconnected such that one category cannot be changed without affecting the others.
This creates an environment where a balance must be established to determine the appropriate amount of effort to focus on each of these three categories.
This current research presents an application for the schedule optimization of operations with flexible durations. The objective of the problem is to find the optimal sequence of jobs to minimize cost, where cost is a function of makespan and work center processing durations. The constraints in the problem include variable job durations and a sequential flow of jobs through multiple successive work centers.
Details of the implemented tools chosen in support of the optimization are described followed by a summary of the accuracy, robustness, and scalability of the proposed system on generic case studies.
Final results indicate the proposed approach can successfully be applied to scheduling optimization with flexible job processing times. Implementation of this approach proves to be accurate, robust, and scalable when evaluated against other approaches. The system is implemented via a user friendly graphical interface.
ACKNOWLEDGMENTS
Throughout my academic experience and writing this thesis, many have offered guidance, all of who deserve thanks. However the encouragement of a select few has led to the completion of this work.
Dr. Manbir Sodhi – your support and guidance throughout my undergraduate and graduate terms. Thank you for presenting me with a problem and allowing me the opportunity to develop my capability of learning. I have no doubt I would not be where I am today if we had never crossed paths and for that, I can never thank you enough.
Arash Nashrollahishirazi – your guidance and advisement truly broadened my academic horizon relating to python and genetic algorithms.
My family – your loving support has helped me through writing the chapters of this work. Without your love and encouragement, none of this would have been possible.
Amgen Inc. - your financial support during my graduate term.
iii
TABLE OF CONTENTS
ABSTRACT ...... ii
ACKNOWLEDGMENTS ...... iii
TABLE OF CONTENTS ...... iv
LIST OF TABLES ...... vi
LIST OF FIGURES ...... vii
CHAPTER
1 Introduction ...... 1
2 Review of Literature ...... 3
2.1 Schedule Optimization ...... 3
2.2 Model Research Applications ...... 4
3 Methodology ...... 5
3.1 Process Simulation and Modeling ...... 5
3.2 Components of Profit ...... 5
3.3 Area of Optimization ...... 8
3.4 P VS. N ...... 9
3.5 Mixed Integer Program Formulation...... 9
3.5.1 Variables and Inputs ...... 10
3.5.2 Objective function ...... 11
3.5.3 Constraints ...... 11
3.6 Optimization Techniques ...... 12
iv
3.7 Algorithm Selection ...... 12
3.8 Algorithm Implementation ...... 13
3.9 System Design ...... 17
3.10 Additional Features ...... 19
3.11 Limitations ...... 20
4 Findings ...... 22
4.1 Case Studies ...... 22
4.1.1 Case Study 1 ...... 22
4.1.2 Case Study 2 ...... 25
4.1.3 Case Study 3 ...... 29
4.1.4 Case Study 4 ...... 31
4.1.5 Case Study 5 ...... 33
5 Conclusion ...... 36
APPENDIX ...... 37
GA PSEUDOCODE ...... 37
BIBLIOGRAPHY ...... 38
v
LIST OF TABLES
TABLE PAGE
Table 1. Case study 1 parameter inputs...... 23
Table 2. Case study 1 method performance comparison...... 25
Table 3. Case study 2 parameter inputs...... 26
Table 4. Case study 2 method performance comparison...... 28
Table 5. Case study 3 parameter inputs...... 29
Table 6. Case study 3 method performance comparison...... 30
Table 7. Case study 4 parameter inputs...... 31
Table 8. Case study 4 method performance comparison...... 32
Table 9. Case study 5 parameter inputs...... 33
Table 10. Case study 5 method performance comparison...... 35
vi
LIST OF FIGURES
FIGURE PAGE
Figure 1. Graphical representation of the problem design...... 2
Figure 2. Correlation of maintenance and equipment usage cycles...... 6
Figure 3. Economic analysis of Boeing commercial airplane maintenance...... 7
Figure 4. Venn diagram showing the relationship between P, NP, and NP-complete ... 9
Figure 5. GA chromosome of work center durations...... 13
Figure 6. Graphical representation of an iteration within example problem...... 14
Figure 7. Crossover process within GA...... 15
Figure 8. Mutation process within GA...... 16
Figure 9. C# GUI interface for providing optimization problem inputs...... 17
Figure 10. C# GUI interface for displaying solution...... 18
Figure 11. Process flow of the C# and python system...... 18
Figure 12. Scheduler graphical display for re-optimization example...... 20
Figure 13. 3D plot presenting the graphical solution for case study 1...... 23
Figure 14. Contour plot presenting the graphical solution for case study 1...... 24
Figure 15. Chart showing divergence to a solution for case study 1...... 24
Figure 16. Scheduler graphical display for case study 1...... 25
Figure 17. 3D plot presenting the graphical solution for case study 2...... 27
Figure 18. Contour plot presenting the graphical solution for case study 2...... 27
Figure 19. Chart showing divergence to a solution for case study 2...... 28
Figure 20. Scheduler graphical display for case study 2...... 28
vii
Figure 21. Chart showing divergence to a solution for case study 3...... 30
Figure 22. Scheduler graphical display for case study 3...... 30
Figure 23. Chart showing divergence to a solution for case study 4...... 32
Figure 24. Scheduler graphical display for case study 4...... 32
Figure 25. Chart showing divergence to a solution for case study 5...... 34
Figure 26. Scheduler graphical display for case study 5 after 250 iterations...... 34
Figure 27. Scheduler graphical display for case study 5 after 1000 iterations...... 35
Figure A.28. Proposed GA pseudocode for the problem. (Nashrollahishirazi, Roy and
Sodhi n.d.)...... 37
viii
CHAPTER 1
INTRODUCTION
When used effectively, schedule optimization can be very valuable to industry.
Two goals strived in manufacturing are to make products less expensive and at a faster production rate because often the customer wants the product as soon as possible and at the lowest price. The customer will likely purchase from a competitor if they feel they can get the product faster or for a lower price. Due to the complexities of manufacturing systems, schedule optimization is computationally expensive and difficult to implement. Some of the complexities of implementing schedule optimization are not considered within the scope of this study including work center downtime, labor capacity, product changeover tasks, interruptible operations, product demand fluctuations, job priorities, and resource utilization.
The design of the problem will include jobs that will be processed through work centers (WCs). There are a total of j jobs to be processed, s work centers in series, and p work centers in parallel. Each job must sequentially go through every serial work center. A job may sit idle between work centers if there is no subsequent work center available. Figure 1 is a graphical illustration of the design of the scheduling optimization problem with flexible durations. Case studies will be evaluated with different quantities of j, s, and p. The components of profit will include functions of makespan and work center process durations. The total profit will be evaluated to determine the quality of the solution.
1
Figure 1. Graphical representation of the problem design.
For manufacturing systems observed by the author, typically there is one order of magnitude quantity of work centers in parallel, one or two orders of magnitude quantity of work centers in series, and anywhere between one to numerous orders of magnitude quantity of jobs. Scheduling optimization becomes increasingly computationally demanding as the size of a manufacturing system increases.
2
CHAPTER 2
REVIEW OF LITERATURE
2.1 Schedule Optimization
There has been much research related to schedule optimization of manufacturing systems. Balasubramanian and Grossmann (2003) used a mixed integer linear programming (MILP) technique to optimize the minimum makespan of tasks with uncertainty in processing time durations. The approach presented by Balasubramanian and Grossmann worked well where the solution was more heavily structured by certain types of constraints but the models did not perform as well when tested against general problems. Li and Ierapetritou (2008) also evaluated schedule optimization with uncertainty in processing time durations but used three formulations of robust optimization (RO). The drawback of the approach taken by Li and Ierapetritou's was that it did not optimize an expected value objective function but rather promised to enforce the feasibility for the entire predefined uncertainty space of the problem. Ant colony optimization was presented by Kumar et. al. (2003) as an effective method of scheduling jobs in a flexible manufacturing system (FMS) but with non-flexible task durations. One of the most popular schedule optimization methods presented in this literature review is genetic algorithm (GA) optimization. Guo et. al. (2008) evaluated
GA optimization with tardiness and earliness penalties built into the objective function which performed effectively when evaluated against two-order scheduling problems.
Gao et. al. (2006) assessed GA optimization with machine availability constraints and
Fanti et. al. (1998) studied GA optimization with a multi-criteria objective function by
3
weighing desired criteria such as makespan, resource utilization, and work in progress
(WIP). To the author’s knowledge, no specific work relating to schedule optimization with flexible processing durations using GA optimization or any other optimization method is published in literature.
2.2 Model Research Applications
There does not seem to be any published research devoted to improving schedule optimization with operations that have flexible durations. This may be due to the fact that scheduling optimization with static work center durations is by itself challenging thus allowing work center durations to be dynamic only increases the difficulty to find the optimal solution. It could also be a consequence of assuming that most of the cost and speed benefits are obtained by solving a scheduling problem with static durations however, under certain circumstances it may be desirable to allow the optimization algorithm to determine task durations to increase profit margins. The remainder of this paper will examine this problem, evaluate the performance of a GA approach against case studies, and conclude with a summary of the results.
4
CHAPTER 3
METHODOLOGY
3.1 Process Simulation and Modeling
Simulations provide the user with a means to better understand a system by allowing the user to conduct experiments and analyze a system without having to deal with the complications inherent with many real world experiments. Simulation package design can range between being tailored for a specific process with specific equipment to having a generic purpose with generic equipment. The advantage to having the more specific simulation package is the developer can spend less time creating the framework for the simulation and more time with the details unique to the system. The disadvantage to having the more specific simulation package is that the user is typically limited to the constraints of the pre-established framework and thus can lose flexibility. In contrast, a generic simulation package generally allows the user more flexibility for development yet typically requires additional time to create the framework. Ideally, an off the shelf simulation package should provide the user adequate customizability to add, remove, or modify features of the simulation with minimal development effort.
3.2 Components of Profit
One component of profit (as defined by this paper) is the cost to operate work centers. Maintenance is a significant input to the cost to operate work centers.
Scheduled equipment maintenance (also known as preventative maintenance) is
5
generally correlated with the quantity of occurrences of using a piece of equipment
(also known as cycles or intervals). Preventative maintenance is typically scheduled prior to the wear zone which is the point where the probability of corrective maintenance (also known as unscheduled maintenance) becomes increasingly
(sometimes exponentially) more likely to occur. Figure 2 shows the general correlation between equipment failure rate and equipment interval, including examples of preventative and corrective maintenance occurrences relative to equipment interval and equipment failure rate.
Figure 2. Correlation of maintenance and equipment usage cycles.
Boeing conducted a statistical analysis to evaluate the economics of conducting maintenance on their commercial airplanes (McLoughlin, Doulatshahi and Onorati
2011). This analysis was used to determine the optimal number of cycles (flight hours) to minimize costs associated with preventative maintenance and corrective maintenance. Figure 3 shows an economic analysis of maintenance to determine the optimal maintenance interval for Boeing commercial airplanes by evaluating the total cost of scheduled and unscheduled maintenance.
6
Figure 3. Economic analysis of Boeing commercial airplane maintenance.
This paper will assume maintenance will occur at the cost optimal quantity of cycles with no direct correlation to time. Given maintenance is correlated to equipment cycles, the cost of maintenance increases per unit of time as the number of cycles increase per that same unit of time. In addition to this maintenance cost, the paper will also assume labor and utilities is a linear addition to the work center cost.
Therefore as a work center production rate increases, so does the cost to run that work center.
Work center cost constitutes half the profit equation whereas the other half comprises of the makespan cost. The components of makespan cost are more indirect than the components of work center cost. The carrying cost of inventory and stock out cost significantly impact the cost of makespan. Some of the components of carrying cost of inventory include taxes, insurance, depreciation, and inventory cost. The cost of inventory is money that is tied up in unfinished product which cannot be used to invest elsewhere until the product is sold (known as cost of capital or the opportunity cost of the money). Therefore, reducing the amount of unfinished product by reducing
7
makespan duration will liquidate inventory assets and allow cash to be invested in something else, such as mutual funds, treasuries, or even money market accounts.
Reducing the amount of unfinished product by reducing makespan duration will also reduce taxes, insurance, and depreciation associated with work-in-progress inventory.
Shorter makespan durations also reduce the risk of long stock out durations because raw materials can be turned into finished product faster. The cost of long stock out durations is a risk based calculation and is correlated with the inventory level of finished goods.
In practice, the cost equations for work centers and for makespan may be a challenge to define and will likely vary between industries and between companies.
However for this paper, the cost equations for work centers and for makespan will be evaluated as quadratic and exponential.
3.3 Area of Optimization
Optimization is an attempt to select the best solution out of a collection of solutions, generally bounded by constraints. One area of optimization is schedule optimization which is an area of operations research that can be used widely among commercial industry. Almost every company has to do some form of scheduling in one way or another. Not every scheduling operation necessarily warrants formulating the problem and solving it using an optimization algorithm but when the problem becomes complex and having a sub-optimal solution may significantly impact cost or speed, then it may be appropriate to leverage schedule optimization.
8
3.4 P VS. N
The types of optimization problems are commonly categorized between P and
NP. P is any type of problem that can be solved in polynomial time and NP is any type of problem that can be solved by a non-deterministic Turing machine in polynomial time (Garey and Johnson 1979). All P problems are NP problems (P ∈ NP) but not all
NP problems are P problems. It takes significantly more time to solve an NP type problem than it does to describe the problem so consequently large NP type problems are typically computationally expensive (Garey and Johnson 1979). There are sub classes for NP type problems including NP-hard, NP-complete, NP-easy, and NP- equivalent. Job-shop schedule optimization typical falls into the NP-complete category
(Garey and Johnson 1979) (Garey, Johnson and Sethi 1976) (Gonzalez and Sahni
1978) (Lenstra, Kan and Brucker 1977). Figure 4 is a Venn diagram showing the relationship between P, NP, and NP-complete.
Figure 4. Venn diagram showing the relationship between P, NP, and NP-complete.
3.5 Mixed Integer Program Formulation
A formulation is presented in the next subsections for the problem of schedule optimization with flexible durations. The minimum degree of complexity of this
9
formulation is quadratic as a consequence of the product of the inputs parameters and decision variables. This is one of many ways to formulate this problem and depending on how the problem is formulated it may impact the appropriate optimization algorithm to use. For example, there may exist a formulation that replaces the input parameters start and end times with durations, such is the case for the GA approach discussed later in the paper.
3.5.1 Variables and Inputs
Sets and indices:
j job index where j 1, 2, 3,…, J
p work center in parallel index where p 1, 2, 3,…, P
s work center in series index where s 1, 2, 3,…, S
Input parameters: